on chain complexes/model structure on cosimplicial abelian groups
related by the Dold-Kan correspondence
on algebras over an operad, on modules over an algebra over an operad
on dendroidal sets, for dendroidal complete Segal spaces, for dendroidal Cartesian fibrations
A category with weak equivalences is an ordinary category with a class of morphisms singled out – called ‘weak equivalences’ – that include the isomorphisms, but also typically other morphisms. Such a category can be thought of as a presentation of an (∞,1)-category that defines explicitly only the 1-morphisms (as opposed to n-morphisms for all $n$) and the information about which of these morphisms should become equivalences in the full (∞,1)-category.
The desired $(\infty,1)$-category in question can be constructed from such a “presentation” by “freely adjoining inverse equivalences” to the weak equivalences, in a suitable $(\infty,1)$-categorical sense. One way to make this precise is by the process of simplicial localization . A single $(\infty,1)$-category can admit many different such presentations. See the section Presentations of (∞,1)-categories below for more details.
A category with weak equivalences is a category $C$ equipped with a subcategory (in the naïve sense) $W \subset C$
which contains all isomorphisms of $C$;
which satisfies two-out-of-three: for $f, g$ any two composable morphisms of $C$, if two of $\{f, g, g \circ f\}$ are in $W$, then so is the third.
Often categories with weak equivalences are equipped with further extra structure that helps with computing the simplicial localization, the homotopy category and derived functors.
In a homotopical category the condition on the weak equivalences is slightly stronger; see below.
In a relative category the condition is slightly weaker. Relative categories have a good homotopy theory. See at relative category.
In a category of fibrant objects there are additional auxiliary morphisms called fibrations.
In a Waldhausen category there are additional auxiliary morphisms called cofibrations.
In a model category there are both of these additional auxiliary classes of morphisms with special interrelation between them.
Other variants include
Three additional conditions which categories with weak equivalences often satisfy are:
the weak equivalences are closed under retracts, as a subcategory of the arrow category of $C$.
the weak equivalences satisfy the two-out-of-six property; in this case the category is called a homotopical category.
the weak equivalences are saturated in the sense that any morphism which becomes an isomorphism in the localization $C[W^{-1}]$ is already a weak equivalence. (This is unrelated to the notion of saturated class of maps used in the theory of weak factorization systems.)
In fact, these three conditions are closely related.
Obviously, saturation implies closure under retracts and two-out-of-six, since the isomorphisms in any category satisfy both.
In any model category, all three conditions hold automatically.
If the weak equivalences admit a calculus of fractions, or a well-behaved class of cofibrations or fibrations, then the three conditions are equivalent. See two-out-of-six property for the proofs, which are from Categories and Sheaves (for the calculus of fractions) and Blumberg-Mandell (for the case of cofibrations, in the context of a Waldhausen category).
If we denote by $Core(C)$ the core of $C$ – the maximal subgroupoid of $C$ – then we have a chain of inclusions $Core(C) \hookrightarrow W \hookrightarrow C$.
Many categories with weak equivalences can be equipped with the further structure of a model category. On the other hand, some categories with weak equivalences can not be equipped with a useful structure of a model category. In particular, categories of diagrams in a model category do not always inherit a useful model structure (on the other hand often they do, see model structure on functors). Several concepts exist that weaken the axioms of a model category in order to still obtain useful results in such a case – for instance a category of fibrant objects.
A category $C$ with weak equivalences serves as a presentation of an (∞,1)-category $\mathbf{C}$ with the same objects and at least the 1-morphisms of $C$, and such that every weak equivalence in $C$ becomes a true equivalence (a homotopy equivalence) in $\mathbf{C}$.
The procedure (or one of its equivalent variants) that constructs the (∞,1)-category $\mathbf{C}$ from the category with weak equivalences $C$ is called Dwyer-Kan simplicial localization.
In fact, every (∞,1)-category may be presented this way (and indeed posets equipped with wide subcategories of morphisms called weak equivalences are sufficient). This is discussed at
Alternatively, we may further project to the 1-category in which all weak equivalences become true isomorphisms: this is the homotopy category of $C$ with respect to $W$. Equivalently this is the homotopy category of an (∞,1)-category of $\mathbf{C}$.
Note that the category with weak equivalences which presents a given $(\infty,1)$-category can not, in general, be taken to be the homotopy category of that $(\infty,1)$-category; more “flab” must be built into it.
It also cannot, in general, be the underlying 1-category of a simplicially enriched presentation of that $(\infty,1)$-category. For instance, every $\infty$-groupoid can be realized as a simplicially enriched groupoid, but the underlying 1-category of a simplicially enriched groupoid is a 1-groupoid, which cannot be localized any further to produce a non-1-truncated $\infty$-groupoid.